Let A be an abelian group. An A-magic of a graph G = (V, E) is a labeling l : E(G) → A \ {0} such that the sum of the labels of the edges incident with u ∈ V is a constant, where 0 is the identity element of the group A. In this paper, we will show that some classes of graphs are A-magic for all abelian group A of even order other than 2. Also, we prove that product and composition of A-magic g...

Given an abelian group A, a graph G = (V, E) is said to have a distance two magic labeling in A if there exists a labeling l : E(G) → A − {0} such that the induced vertex labeling l∗ : V (G) → A defined by l∗(v) = ∑ e∈E(v) l(e) is a constant map, where E(v) = {e ∈ E(G) : d(v, e) < 2}. The set of all h ∈ Z+ for which G has a distance two magic labeling in Zh is called the distance two magic spec...

Infinite nuclear matter mass model and the relativistic mean field theory show strong evidence of new neutron magic numbers 100, 150, 164; proton magic number 78 and new islands of stability around N=100, Z ≃ 62; N=150, Z=78; and N= 164, Z ≃ 90 in the drip-line regions of nuclear chart. It is shown that shell effect stabilizes here the instability due to nuclear force giving rise to a new pheno...

Let F ={H1, . . . ,Hk} be a family of graphs. A graph G is called totally F -decomposable iffor every linear combination of the form α1e(H1) + · · · + αke(Hk) = e(G) where each αi is anonnegative integer, there is a coloring of the edges of G with α1 + · · · + αk colors such thatexactly αi color classes induce each a copy of Hi, for i = 1, . . . , k. We prove that if F i...

Alspach conjectured that every connected Cayley graph on a finite Abelian group A is Hamiltondecomposable. Liu has shown that for |A| even, if S = {s1, . . . , sk} ⊂ A is an inverse-free strongly minimal generating set of A, then the Cayley graph Cay(A;S?), is decomposable into k Hamilton cycles, where S? denotes the inverse-closure of S. Extending these techniques and restricting to the 6-regu...

We introduce Clique Matrices as an alternative representation of undirected graphs, being a generalisation of the incidence matrix representation. Here we use clique matrices to decompose a graph into a set of possibly overlapping clusters, defined as wellconnected subsets of vertices. The decomposition is based on a statistical description which encourages clusters to be well connected and few...

We make several contributions in accelerating approximate Bayesian structural inference for non-decomposable GGMs. Our first contribution is to show how to efficiently compute a BIC or Laplace approximation to the marginal likelihood of non-decomposable graphs using convex methods for precision matrix estimation. This optimization technique can be used as a fast scoring function inside standard...

A k-container C(u, v) of a graph G is a set of k-disjoint paths joining u to v. A k-container C(u, v) of G is a k∗-container if it contains all the vertices of G. A graph G is k∗-connected if there exists a k∗-container between any two distinct vertices. Let (G) be the connectivity of G. A graph G is super connected if G is i∗-connected for all 1 i (G). A bipartite graph G is k∗-laceable if the...

Modern approaches for optimally learning Bayesian network structures require decomposable scores. Such approaches include those based on dynamic programming and heuristic search methods. These approaches operate in a search space called the order graph, which has been investigated extensively in recent years. In this paper, we break from this tradition, and show that one can effectively learn s...